3.195 \(\int \frac{\text{sech}^5(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=89 \[ \frac{3 \tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac{\tanh \left (a+b \log \left (c x^n\right )\right ) \text{sech}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \tanh \left (a+b \log \left (c x^n\right )\right ) \text{sech}\left (a+b \log \left (c x^n\right )\right )}{8 b n} \]

[Out]

(3*ArcTan[Sinh[a + b*Log[c*x^n]]])/(8*b*n) + (3*Sech[a + b*Log[c*x^n]]*Tanh[a + b*Log[c*x^n]])/(8*b*n) + (Sech
[a + b*Log[c*x^n]]^3*Tanh[a + b*Log[c*x^n]])/(4*b*n)

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Rubi [A]  time = 0.0571088, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3768, 3770} \[ \frac{3 \tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac{\tanh \left (a+b \log \left (c x^n\right )\right ) \text{sech}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \tanh \left (a+b \log \left (c x^n\right )\right ) \text{sech}\left (a+b \log \left (c x^n\right )\right )}{8 b n} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*Log[c*x^n]]^5/x,x]

[Out]

(3*ArcTan[Sinh[a + b*Log[c*x^n]]])/(8*b*n) + (3*Sech[a + b*Log[c*x^n]]*Tanh[a + b*Log[c*x^n]])/(8*b*n) + (Sech
[a + b*Log[c*x^n]]^3*Tanh[a + b*Log[c*x^n]])/(4*b*n)

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\text{sech}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \text{sech}^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\text{sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \operatorname{Subst}\left (\int \text{sech}^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n}\\ &=\frac{3 \text{sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{\text{sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \operatorname{Subst}\left (\int \text{sech}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{8 n}\\ &=\frac{3 \tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac{3 \text{sech}\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{\text{sech}^3\left (a+b \log \left (c x^n\right )\right ) \tanh \left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end{align*}

Mathematica [A]  time = 0.0891788, size = 75, normalized size = 0.84 \[ \frac{3 \tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )+2 \tanh \left (a+b \log \left (c x^n\right )\right ) \text{sech}^3\left (a+b \log \left (c x^n\right )\right )+3 \tanh \left (a+b \log \left (c x^n\right )\right ) \text{sech}\left (a+b \log \left (c x^n\right )\right )}{8 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + b*Log[c*x^n]]^5/x,x]

[Out]

(3*ArcTan[Sinh[a + b*Log[c*x^n]]] + 3*Sech[a + b*Log[c*x^n]]*Tanh[a + b*Log[c*x^n]] + 2*Sech[a + b*Log[c*x^n]]
^3*Tanh[a + b*Log[c*x^n]])/(8*b*n)

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Maple [A]  time = 0.022, size = 84, normalized size = 0.9 \begin{align*}{\frac{ \left ({\rm sech} \left (a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}\tanh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{4\,bn}}+{\frac{3\,{\rm sech} \left (a+b\ln \left ( c{x}^{n} \right ) \right )\tanh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{8\,bn}}+{\frac{3\,\arctan \left ({{\rm e}^{a+b\ln \left ( c{x}^{n} \right ) }} \right ) }{4\,bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(a+b*ln(c*x^n))^5/x,x)

[Out]

1/4*sech(a+b*ln(c*x^n))^3*tanh(a+b*ln(c*x^n))/b/n+3/8*sech(a+b*ln(c*x^n))*tanh(a+b*ln(c*x^n))/b/n+3/4/b/n*arct
an(exp(a+b*ln(c*x^n)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 96 \, c^{b} \int \frac{e^{\left (b \log \left (x^{n}\right ) + a\right )}}{128 \,{\left (c^{2 \, b} x e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + x\right )}}\,{d x} + \frac{3 \, c^{7 \, b} e^{\left (7 \, b \log \left (x^{n}\right ) + 7 \, a\right )} + 11 \, c^{5 \, b} e^{\left (5 \, b \log \left (x^{n}\right ) + 5 \, a\right )} - 11 \, c^{3 \, b} e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} - 3 \, c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )}}{4 \,{\left (b c^{8 \, b} n e^{\left (8 \, b \log \left (x^{n}\right ) + 8 \, a\right )} + 4 \, b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 6 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 4 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(a+b*log(c*x^n))^5/x,x, algorithm="maxima")

[Out]

96*c^b*integrate(1/128*e^(b*log(x^n) + a)/(c^(2*b)*x*e^(2*b*log(x^n) + 2*a) + x), x) + 1/4*(3*c^(7*b)*e^(7*b*l
og(x^n) + 7*a) + 11*c^(5*b)*e^(5*b*log(x^n) + 5*a) - 11*c^(3*b)*e^(3*b*log(x^n) + 3*a) - 3*c^b*e^(b*log(x^n) +
a))/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) + 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*log(x^n
) + 4*a) + 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n)

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Fricas [B]  time = 3.31076, size = 4298, normalized size = 48.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(a+b*log(c*x^n))^5/x,x, algorithm="fricas")

[Out]

1/4*(3*cosh(b*n*log(x) + b*log(c) + a)^7 + 21*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^
6 + 3*sinh(b*n*log(x) + b*log(c) + a)^7 + (63*cosh(b*n*log(x) + b*log(c) + a)^2 + 11)*sinh(b*n*log(x) + b*log(
c) + a)^5 + 11*cosh(b*n*log(x) + b*log(c) + a)^5 + 5*(21*cosh(b*n*log(x) + b*log(c) + a)^3 + 11*cosh(b*n*log(x
) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^4 + (105*cosh(b*n*log(x) + b*log(c) + a)^4 + 110*cosh(b*n*l
og(x) + b*log(c) + a)^2 - 11)*sinh(b*n*log(x) + b*log(c) + a)^3 - 11*cosh(b*n*log(x) + b*log(c) + a)^3 + (63*c
osh(b*n*log(x) + b*log(c) + a)^5 + 110*cosh(b*n*log(x) + b*log(c) + a)^3 - 33*cosh(b*n*log(x) + b*log(c) + a))
*sinh(b*n*log(x) + b*log(c) + a)^2 + 3*(cosh(b*n*log(x) + b*log(c) + a)^8 + 8*cosh(b*n*log(x) + b*log(c) + a)*
sinh(b*n*log(x) + b*log(c) + a)^7 + sinh(b*n*log(x) + b*log(c) + a)^8 + 4*(7*cosh(b*n*log(x) + b*log(c) + a)^2
+ 1)*sinh(b*n*log(x) + b*log(c) + a)^6 + 4*cosh(b*n*log(x) + b*log(c) + a)^6 + 8*(7*cosh(b*n*log(x) + b*log(c
) + a)^3 + 3*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^5 + 2*(35*cosh(b*n*log(x) + b*lo
g(c) + a)^4 + 30*cosh(b*n*log(x) + b*log(c) + a)^2 + 3)*sinh(b*n*log(x) + b*log(c) + a)^4 + 6*cosh(b*n*log(x)
+ b*log(c) + a)^4 + 8*(7*cosh(b*n*log(x) + b*log(c) + a)^5 + 10*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*cosh(b*n
*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 + 4*(7*cosh(b*n*log(x) + b*log(c) + a)^6 + 15*cosh(
b*n*log(x) + b*log(c) + a)^4 + 9*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(x) + b*log(c) + a)^2 + 4*
cosh(b*n*log(x) + b*log(c) + a)^2 + 8*(cosh(b*n*log(x) + b*log(c) + a)^7 + 3*cosh(b*n*log(x) + b*log(c) + a)^5
+ 3*cosh(b*n*log(x) + b*log(c) + a)^3 + cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)
*arctan(cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)) + (21*cosh(b*n*log(x) + b*log(c) +
a)^6 + 55*cosh(b*n*log(x) + b*log(c) + a)^4 - 33*cosh(b*n*log(x) + b*log(c) + a)^2 - 3)*sinh(b*n*log(x) + b*lo
g(c) + a) - 3*cosh(b*n*log(x) + b*log(c) + a))/(b*n*cosh(b*n*log(x) + b*log(c) + a)^8 + 8*b*n*cosh(b*n*log(x)
+ b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^7 + b*n*sinh(b*n*log(x) + b*log(c) + a)^8 + 4*b*n*cosh(b*n*log
(x) + b*log(c) + a)^6 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + b*n)*sinh(b*n*log(x) + b*log(c) + a)^6 +
6*b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*b*n*cosh(b*n*log(x) +
b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^5 + 2*(35*b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 30*b*n*cosh(
b*n*log(x) + b*log(c) + a)^2 + 3*b*n)*sinh(b*n*log(x) + b*log(c) + a)^4 + 4*b*n*cosh(b*n*log(x) + b*log(c) + a
)^2 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^5 + 10*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*b*n*cosh(b*n*l
og(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^6 + 15*b*n
*cosh(b*n*log(x) + b*log(c) + a)^4 + 9*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + b*n)*sinh(b*n*log(x) + b*log(c)
+ a)^2 + b*n + 8*(b*n*cosh(b*n*log(x) + b*log(c) + a)^7 + 3*b*n*cosh(b*n*log(x) + b*log(c) + a)^5 + 3*b*n*cos
h(b*n*log(x) + b*log(c) + a)^3 + b*n*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{5}{\left (a + b \log{\left (c x^{n} \right )} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(a+b*ln(c*x**n))**5/x,x)

[Out]

Integral(sech(a + b*log(c*x**n))**5/x, x)

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Giac [A]  time = 1.17997, size = 205, normalized size = 2.3 \begin{align*} \frac{1}{4} \, c^{5 \, b}{\left (\frac{3 \, \arctan \left (\frac{c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right ) e^{\left (-5 \, a\right )}}{b c^{4 \, b} c^{b} n} + \frac{{\left (3 \, c^{6 \, b} x^{7 \, b n} e^{\left (6 \, a\right )} + 11 \, c^{4 \, b} x^{5 \, b n} e^{\left (4 \, a\right )} - 11 \, c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} - 3 \, x^{b n}\right )} e^{\left (-4 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{4} b c^{4 \, b} n}\right )} e^{\left (5 \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(a+b*log(c*x^n))^5/x,x, algorithm="giac")

[Out]

1/4*c^(5*b)*(3*arctan(c^(2*b)*x^(b*n)*e^a/c^b)*e^(-5*a)/(b*c^(4*b)*c^b*n) + (3*c^(6*b)*x^(7*b*n)*e^(6*a) + 11*
c^(4*b)*x^(5*b*n)*e^(4*a) - 11*c^(2*b)*x^(3*b*n)*e^(2*a) - 3*x^(b*n))*e^(-4*a)/((c^(2*b)*x^(2*b*n)*e^(2*a) + 1
)^4*b*c^(4*b)*n))*e^(5*a)